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Unanswered: CUMULATIVE COUNTIF...sorta

I want to have a cumulative COUNTIFS. I want it to countif "criteria 1" is there and then add another one to the count if "criteria 2" is present and add another one to the count if criteria 4 is present so in the end I can get a total of 3 in the cell and not one.

Now lets say I have a category column titled Teachers and I wanted to count how many teachers where in that cell.